Robust D-stability analysis via polynomial positivity

This paper is concerned with robust D-stability of linear systems depending polynomially on uncertain parameters which belong to semi-algebraic sets. The robust stability condition is converted into checking whether a polynomial is positive over a semi algebraic set. Based on sum-of-squares relaxations, a sufficient condition for the polynomial positivity can be formulated as solving a linear matrix inequality (LMI). Construction of a hierarchy of the LMI relaxations, which converge to the stability condition, is also possible via the degree increase of the polynomial. Moreover, a condition to verify instability amounts to solving polynomial equations and inequalities, whose LMI relaxations are available.